Does the Earth rise and set on the Moon?
No: from the near side the Earth hangs almost still, only wobbling a little over a month.
The Moon turns once on its axis in exactly the time it takes to go round the Earth, so the same side always faces us — and from that side the Earth always stays in the same part of the sky.
It does wobble a few degrees (libration), so near the edge of the near side the Earth bobs along the horizon. From the far side it is never seen at all.
What does change is the Earth's phase: when we see a thin crescent Moon, the Moon sees an almost full Earth.